Schaum's Outline of Differential Equations, 3rd edition

by Bronson, Richard

• ISBN13: 9780071456876
• ISBN10: 0071456872
• Edition: 2nd
• Format: Paperback
• Copyright: 2006-06-15
• Publisher: McGraw-Hill

Summary

Solve differential equations faster and easier! Thoroughly updated, this third edition of "Schaum's Outline of Differential Equations" offers you new, faster techniques for solving differential equations generated by the emergence of high-speed computers. Differential equations, a linchpin of modern math, are essential in engineering, the natural sciences, economics, and business. Includes: 563 fully solved problems 800-plus supplementary problems New chapter on modeling

Author Biography

Richard Bronson, Ph.D., is a professor of mathematics at Fairleigh Dickinson University.

Gabriel Costa, Ph.D., is an associate professor of mathematics in the Department of Mathematical Sciences at West Point.

Table of Contents

Basic Concepts	1	(8)
Differential Equations	1	(1)
Notation	2	(1)
Solutions	2	(1)
Initial-Value and Boundary-Value Problems	2	(7)
An Introduction to Modeling and Qualitative Methods	9	(5)
Mathematical Models	9	(1)
The ``Modeling Cycle''	9	(1)
Qualitative Methods	10	(4)
Classifications of First-Order Differential Equations	14	(7)
Standard Form and Differential Form	14	(1)
Linear Equations	14	(1)
Bernoulli Equations	14	(1)
Homogeneous Equations	15	(1)
Separable Equations	15	(1)
Exact Equations	15	(6)
Separable First-Order Differential Equations	21	(10)
General Solution	21	(1)
Solutions to the Initial-Value Problem	21	(1)
Reduction of Homogeneous Equations	22	(9)
Exact First-Order Differential Equations	31	(11)
Defining Properties	31	(1)
Method of Solution	31	(1)
Integrating Factors	32	(10)
Linear First-Order Differential Equations	42	(8)
Method of Solution	42	(1)
Reduction of Bernoulli Equations	42	(8)
Applications of First-Order Differential Equations	50	(23)
Growth and Decay Problems	50	(1)
Temperature Problems	50	(1)
Falling Body Problems	51	(1)
Dilution Problems	52	(1)
Electrical Circuits	52	(1)
Orthogonal Trajectories	53	(20)
Linear Differential Equations: Theory of Solutions	73	(10)
Linear Differential Equations	73	(1)
Linearly Independent Solutions	74	(1)
The Wronskian	74	(1)
Nonhomogeneous Equations	74	(9)
Second-Order Linear Homogeneous Differential Equations with Constant Coefficients	83	(6)
Introductory Remark	83	(1)
The Characteristic Equation	83	(1)
The General Solution	84	(5)
nth-Order Linear Homogeneous Differential Equations with Constant Coefficients	89	(5)
The Characteristic Equation	89	(1)
The General Solution	90	(4)
The Method of Undetermined Coefficients	94	(9)
Simple Form of the Method	94	(1)
Generalizations	95	(1)
Modifications	95	(1)
Limitations of the Method	95	(8)
Variation of Parameters	103	(7)
The Method	103	(1)
Scope of the Method	104	(6)
Initial-Value Problems for Linear Differential Equations	110	(4)
Applications of Second-Order Linear Differential Equations	114	(17)
Spring Problems	114	(1)
Electrical Circuit Problems	115	(1)
Buoyancy Problems	116	(1)
Classifying Solutions	117	(14)
Matrices	131	(9)
Matrices and Vectors	131	(1)
Matrix Addition	131	(1)
Scalar and Matrix Multiplication	132	(1)
Powers of a Square Matrix	132	(1)
Differentiation and Integration of Matrices	132	(1)
The Characteristic Equation	133	(7)
eAt	140	(8)
Definition	140	(1)
Computation of eAt	140	(8)
Reduction of Linear Differential Equations to a System of First-Order Equations	148	(9)
An Example	148	(1)
Reduction of an nth Order Equation	149	(1)
Reduction of a System	150	(7)
Graphical and Numerical Methods for Solving First-Order Differential Equations	157	(19)
Qualitative Methods	157	(1)
Direction Fields	157	(1)
Euler's Method	158	(1)
Stability	158	(18)
Further Numerical Methods for Solving First-Order Differential Equations	176	(19)
General Remarks	176	(1)
Modified Euler's Method	177	(1)
Runge--Kutta Method	177	(1)
Adams--Bashford--Moulton Method	177	(1)
Milne's Method	177	(1)
Starting Values	178	(1)
Order of a Numerical Method	178	(17)
Numerical Methods for Solving Second-Order Differential Equations Via Systems	195	(16)
Second-Order Differential Equations	195	(1)
Euler's Method	196	(1)
Runge--Kutta Method	196	(1)
Adams--Bashford--Moulton Method	196	(15)
The Laplace Transform	211	(13)
Definition	211	(1)
Properties of Laplace Transforms	211	(1)
Functions of Other Independent Variables	212	(12)
Inverse Laplace Transforms	224	(9)
Definition	224	(1)
Manipulating Denominators	224	(1)
Manipulating Numerators	225	(8)
Convolutions and the Unit Step Function	233	(9)
Convolutions	233	(1)
Unit Step Function	233	(1)
Translations	234	(8)
Solutions of Linear Differential Equations with Constant Coefficients by Laplace Transforms	242	(7)
Laplace Transforms of Derivatives	242	(1)
Solutions of Differential Equations	243	(6)
Solutions of Linear Systems by Laplace Transforms	249	(5)
The Method	249	(5)
Solutions of Linear Differential Equations with Constant Coefficients by Matrix Methods	254	(8)
Solution of the Initial-Value Problem	254	(1)
Solution with No Initial Conditions	255	(7)
Power Series Solutions of Linear Differential Equations with Variable Coefficients	262	(13)
Second-Order Equations	262	(1)
Analytic Functions and Ordinary Points	262	(1)
Solutions Around the Origin of Homogeneous Equations	263	(1)
Solutions Around the Origin of Nonhomogeneous Equations	263	(1)
Initial-Value Problems	264	(1)
Solutions Around Other Points	264	(11)
Series Solutions Near a Regular Singular Point	275	(15)
Regular Singular Points	275	(1)
Method of Frobenius	275	(1)
General Solution	276	(14)
Some Classical Different Equations	290	(5)
Classical Differential Equations	290	(1)
Polynomial Solutions and Associated Concepts	290	(5)
Gamma and Bessel Functions	295	(9)
Gamma Function	295	(1)
Bessel Functions	295	(1)
Algebraic Operations on Infinite Series	296	(8)
An Introduction to Partial Differential Equations	304	(5)
Introductory Concepts	304	(1)
Solutions and Solution Techniques	305	(4)
Second-Order Boundary-Value Problems	309	(9)
Standard Form	309	(1)
Solutions	310	(1)
Eigenvalue Problems	310	(1)
Sturm--Liouville Problems	310	(1)
Properties of Sturm--Liouville Problems	310	(8)
Eigenfunction Expansions	318	(7)
Piecewise Smooth Functions	318	(1)
Fourier Sine Series	319	(1)
Fourier Cosine Series	319	(6)
An Introduction to Difference Equations	325	(5)
Introduction	325	(1)
Classifications	325	(1)
Solutions	326	(4)
APPENDIX A Laplace Transforms	330	(6)
APPENDIX B Some Comments about Technology	336	(2)
Introductory Remarks	336	(1)
T1-89	337	(1)
Mathematica	337	(1)
Answers to Supplementary Problems	338	(44)
Index	382